Numerical bifurcation analysis and pattern formation in a minimal reaction-diffusion model for vegetation. (English) Zbl 1483.92155
Summary: Model-aided understanding of the mechanism of vegetation patterns and desertification is one of the burning issues in the management of sustainable ecosystems. A pioneering model of vegetation patterns was proposed by C. A. Klausmeier [“Regular and irregular patterns in semiarid vegetation”, Science 284, No. 5421, 1826–1828 (1999; doi:10.1126/science.284.5421.1826)] that involves a downhill flow of water. In this paper, we study the diffusive Klausmeier model that can describe the flow of water in flat terrain incorporating a diffusive flow of water. It consists of a two-component reaction-diffusion system for water and plant biomass. The paper presents a numerical bifurcation analysis of stationary solutions of the diffusive Klausmeier model extensively. We numerically investigate the occurrence of diffusion-driven instability and how this depends on the parameters of the model. Finally, the model predicts some field observed vegetation patterns in a semiarid environment, e.g. spot, stripe (labyrinth), and gap patterns in the transitions from bare soil at low precipitation to homogeneous vegetation at high precipitation. Furthermore, we introduce a two-component reaction-diffusion model considering a bilinear interaction of plant and water instead of their cubic interaction. It is inspected that no diffusion-driven instability occurs as if vegetation patterns can be generated. This confirms that the diffusive Klausmeier model is the minimal reaction-diffusion model for the occurrence of vegetation patterns from the viewpoint of a two-component reaction-diffusion system.
MSC:
| 92D40 | Ecology |
| 35K57 | Reaction-diffusion equations |
| 34K21 | Stationary solutions of functional-differential equations |
| 35B36 | Pattern formations in context of PDEs |
Keywords:
Klausmeier model; reaction-diffusion system; numerical bifurcation analysis; vegetation patterns; minimal modelReferences:
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